Example 2.4.3.10 (00KX): $2$-Simplices of the Homotopy Coherent Nerve

Example 2.4.3.10 ($2$-Simplices of the Homotopy Coherent Nerve). Let $Q = \{ x_0 < x_1 < x_2 \} $ be a linearly ordered set with three elements. Then the map $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ is not an equivalence of simplicial categories. In the underlying category $\operatorname{Path}[Q]$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ & x_1 \ar [dr]^{ \{ x_1 < x_2 \} } & \\ x_0 \ar [ur]^{ \{ x_0 < x_1 \} } \ar [rr]_{ \{ x_0 < x_2 \} } & & x_2 } \]

does not commute: the composition of the diagonal maps is the path $\{ x_0 < x_1 < x_2 \} $. However, it commutes in a weak sense: there is an edge of the simplicial set $\operatorname{Hom}_{\operatorname{Path}[Q]}(x_0,x_2)_{\bullet }$ having source $\{ x_0 < x_1 < x_2 \} $ and target $\{ x_0 < x_2 \} $. It follows that for any simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, a choice of $2$-simplex

\[ \sigma \in \operatorname{N}_{2}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) = \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[2]_{\bullet } , \operatorname{\mathcal{C}}_{\bullet }) \simeq \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[Q]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) \]

determines a (possibly non-commutative) diagram $\sigma _0$:

\[ \xymatrix@R =50pt@C=50pt{ & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]^{ f_{20} } & & X_2, } \]

in $\operatorname{\mathcal{C}}$, together with a homotopy $h$ from $f_{21} \circ f_{10}$ to $f_{20}$ (in the sense of Definition 2.4.1.6). Conversely, every choice of homotopy from $f_{21} \circ f_{10}$ to $f_{20}$ determines a unique $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ (see Proposition 2.4.6.10).